Credit Risk Modeling

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# Credit Risk Modeling - PowerPoint PPT Presentation

Economic Models of Credit Risk Lectures 10 &amp; 11. Credit Risk Modeling. Based on Risk Management, Crouhy, Galai, Mark, McGraw-Hill, 2000. (Kealhofer / McQuown / Vasicek). The Contingent Claim Approach - Structural Approach: KMV . The Option Pricing Approach: KMV.

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Presentation Transcript
Economic Models of Credit Risk

Lectures 10 & 11

### Credit Risk Modeling

Based on Risk Management, Crouhy, Galai, Mark,

McGraw-Hill, 2000

(Kealhofer / McQuown / Vasicek)

### The Contingent Claim Approach - Structural Approach: KMV

The Option Pricing Approach: KMV

KMV challenges CreditMetrics on several fronts:

1. Firms within the same rating class have the same default rate

2. The actual default rate (migration probabilities) are equal to the historical default rate (migration frequencies)

• Default rates change continuously while ratings are adjusted in a discrete fashion.
• Default rates vary with current economic and financial conditions of the firm.
The Option Pricing Approach: KMV

KMV challenges CreditMetrics on several fronts:

3. Default is only defined in a statistical sense without explicit reference to the process which leads to default.

• KMV proposes a structural model which relates default to balance sheet dynamics
• Microeconomic approach to default: a firm is in default when it cannot meet its financial obligations
• This happens when the value of the firm’s assets falls below some critical level
The Option Pricing Approach: KMV

KMV’s model is based on the option pricing approach to credit risk as originated by Merton (1974)

1. The firm’s asset value follows a standard geometric Brownian motion, i.e.:

dV

=

m

+

s

t

dt

dZ

t

V

t

ì

ü

s

2

=

m

-

+

s

Z

V

V

exp

(

)

t

t

í

ý

0

t

t

2

î

þ

Assets

Liabilities / Equity

Debt: Bt

(F)

Risky Assets: Vt

Equity: St

V

V

Total:

t

t

The Option Pricing Approach: KMV

2. Balance sheet of Merton’s firm

The Option Pricing Approach: KMV

Equity value at maturity of debt obligation:

(

)

=

-

S

max

V

F

,

0

T

T

Firm defaults if

<

V

F

T

with probability of default (“real world” probability measure)

æ

ö

æ

ö

s

2

V

ç

÷

+

ç

m

-

÷

Ln

T

0

ç

÷

F

2

ç

÷

è

ø

(

)

(

)

<

=

Z

<

-

=

-

0

P

V

F

P

N

d

ç

÷

T

T

2

s

T

ç

÷

ç

÷

è

ø

ì

í

î

The Option Pricing Approach: KMV

3. Probability of default (“real world” probability measure)

• Distribution of asset values at maturity of the debt obligation

Assets Value

ü

æ

ö

s

2

=

m

-

+

s

Z

ç

÷

ý

V

V

exp

T

T

T

O

è

ø

T

2

þ

m

=

T

E

(

)

V

e

V

O

T

V

T

V

0

F

Probability of default

Time

T

-

rT

B

+ P

= Fe

0

0

The Option Pricing Approach: KMV

Bank’s pay-off matrix at times 0 and T for making a loan to Firm ABC and buying a put on the value of ABC

Time

0

T

£

Value of Assets

V

V

F

V

F

>

0

T

T

Bank’s Position:

·

-B

F

V

make a loan

0

T

·

-P

F - V

O

0

T

Total

-B

-P

F

F

0

0

Corporate loan = Treasury bond + short a put

• Bo = Fe-rT - Po
• So = Vo - Bo (assuming markets are frictionless)
• Bo = Fe-YTTwhere YT is yield to maturity
• Probability of Default = g (Vo, F, sv, r, T) = N ( - d2 )
• (“risk neutral” probability measure)

6Conditional recovery when default = VT

KMV: Merton’s Model

Firm ABC is structured as follows:

Vt = Value of Assets (at time t)

St = Value of Equity

Bt = Value of Debt (zero-coupon)

F = Face Value of Debt

KMV: Merton’s Model

Problem:

Vo ( say =100 ), F ( say = 77 ), sv ( say = 40% ),

r ( say =10% ) and T ( say = 1 year)

Solve for Bo,So,YT and Probability of Default

KMV: Merton’s Model

`

Solution:

P0( = 3.37) ® Bo( = 66.63) ® So( = 33.37) ® YT ( =15.6%) ®PT ( = 5.6%)

æ

ö

F

=

®

P

=

-

ç

÷

=

-

=

-

Y

L

Y

r

=

-

rT

P

f

(

V

,

T

)

S

V

B

B

Fe

P

K

T

N

T

T

o

o

o

o

o

o

o

è

ø

B

o

Note: In solving for P0 we get Probability of Default ( = 24.4% )

The Option Pricing Approach: KMV

P

=

-

Default spread ( ) for corporate debt

( For V0 = 100, T = 1, and r = 10% )

Y

r

T

T

s

0.05

0.10

0.20

0.40

LR

0.5

0

0

0

1.0

0.6

0

0

0.1%

2.5%

0.7

0

0

0.4%

5.6%

0.8

0

0.1%

1.5%

8.4%

0.9

0.1%

0.8%

4.1%

12.5%

1.0

2.1%

3.1%

8.3%

17.3%

-

rT

Fe

Leverage ratio:

=

LR

V

0

KMV: EDFs (Expected Default Frequencies)

4. Default point and distance to default

Observation:

Firms more likely to default when their asset values reach a certain level of total liabilities and value of short-term debt.

Default point is defined as

DPT=STD+0.5LTD

STD-short-term debt

LTD- long-term debt

KMV: EDFs (Expected Default Frequencies)

Default point (DPT)

Probability distribution of V

Asset Value

Expected growth of

assets, net

E(V)

1

V0

DD

DPT = STD + ½ LTD

Time

1 year

0

KMV: EDFs (Expected Default Frequencies)

Distance-to-default (DD)

DD- is the distance between the expected asset value in T years, E(VT) , and the default point, DPT, expressed in standard deviation of future asset returns:

KMV: EDFs (Expected Default Frequencies)

5. Derivation of the probabilities of default from the distance to default

EDF

40 bp

4

6

1

2

5

3

DD

KMV also uses historical data to compute EDFs

-

1

,

200

800

=

=

DD

4

100

KMV: EDFs (Expected Default Frequencies)

Example:

V0 = 1,000

Current market value of assets:

Net expected growth of assets per annum:

Expected asset value in one year:

Annualized asset volatility,

Default point

20%

:

V1 = V0(1.20) = 1,200

sA

100

800

Assume that among the population of all the firms with DD of 4 at one point in time, e.g. 5,000, 20 defaulted one year later, then:

20

=

=

=

EDF

0

.

004

0

.

4%

or 40 bp

1

year

5

,

000

The implied rating for this probability of default is BB+

KMV: EDFs (Expected Default Frequencies)

Example:Federal Express (\$ figures are in billions of US\$)

November 1997

February 1998

Market capitalization (S0 ) (price* shares outstanding)

Book liabilities

Market value of assets (V0 )

Asset volatility

Default point

Distance to default (DD)

EDF

\$ 7.8

\$ 4.8

\$ 12.6

15%

\$ 3.4

12.6-3.4

0.15·12.6

0.06%(6bp)

\$ 7.3

\$4.9

\$ 12.2

17%

\$ 3.5

12.2-3.5

0.17·12.2

0.11%(11bp)

= 4.9

= 4.2

º A

º AA

KMV: EDFs (Expected Default Frequencies)

4. EDF as a predictor of default

EDF of a firm which actually defaulted versus EDFs of firms in various quartiles and the lower decile.

The quartiles and decile represent a range of EDFs for a specific credit class.

KMV: EDFs (Expected Default Frequencies)

4. EDF as a predictor of default

EDF of a firm which actually defaulted versus Standard & Poor’s rating.

KMV: EDFs (Expected Default Frequencies)

4. EDF as a predictor of default

Assets value, equity value, short term debt and long term debt of a firm which actually defaulted.

Credit Suisse Financial Products

### IV The Actuarial Approach: CreditRisk+

The Actuarial Approach: CreditRisk+

In CreditRisk+ no assumption is made about the causes of default: an obligor A is either in default with probability PA, or it is not in default with probability 1-PA. It is assumed that:

• for a loan, the probability of default in a given period, say one month, is the same for any other month
• for a large number of obligors, the probability of default by any particular obligor is small and the number of defaults that occur in any given period is independent of the number, of defaults that occur in any other period
The Actuarial Approach: CreditRisk+

Under those circumstances, the probability distribution for the number of defaults, during a given period of time (say one year) is well represented by a Poisson distribution:

where

m

= average number of defaults per year

æ

ö

å

m

m

=

P

It is shown that can be approximated as

ç

÷

A

è

ø

A

CreditRisk+: Frequency of default events

One year default rate

Credit Rating

Average (%)

Standard deviation (%)

Aaa

0.00

0.0

Aa

0.03

0.1

A

0.01

0.0

Baa

0.13

0.3

Ba

1.42

1.3

B

7.62

5.1

m

Note, that standard deviation of a Poisson distribution is .

For instance, for rating B: .

m

=

=

7

.

62

2

.

76

versus

5

.

1

CreditRisk+ assumes that default rate is random and has Gamma

distribution with given mean and standard deviation.

Source: Carty and Lieberman (1996)

CreditRisk+: Frequency of default events

Probability

Excluding default rate volatility

Including default rate volatility

Number of defaults

Source: CreditRisk+

Distribution of default events

CreditRisk+: Loss distribution
• In CreditRisk+, the exposure for each obligor is adjusted by the anticipated recovery rate in order to produce a loss given default (exogenous to the model)
CreditRisk+: Loss distribution

1. Losses (exposures, net of recovery) are divided into bands, with the level of exposure in each band being approximated by a single number.

Notation

A

Obligor

Exposure (net of recovery)

LA

PA

Probability of default

lA=LAxPA

Expected loss

CreditRisk+: Loss distribution

Example: 500 obligors with exposures between \$50,000 and \$1M (6 obligors are shown in the table)

Round-offexposure(in \$100,000)

Exposure (\$)(loss given default)

Exposure(in \$100,000)

Obligor

Band

L

n

n

A

j

j

j

A

1

150,000

1.5

2

2

2

460,000

4.6

5

5

3

435,000

4.35

5

5

4

370,000

3.7

4

4

5

190,000

1.9

2

2

6

480,000

4.8

5

5

The unit of exposure is assumed to be L=\$100,000. Each band j, j=1, …, m, with m=10, has an average common exposure: vj=\$100,000j

CreditRisk+: Loss distribution

In Credit Risk+ each band is viewed as an independent

portfolio of loans/bonds, for which we introduce the

following notation:

Notation

Common exposure in band j in units of L nj

nj = \$100,000, \$200,000, …, \$1M

Expected loss in band j in units of L ej

(for all obligors in band j)

Expected number of defaults in band j mj

ej = nj x mj

mj can be expressed in terms of the individual loan characteristics

CreditRisk+: Loss distribution

Number

Band:

of

e

m

j

j

j

obligors

1

30

1.5 (1.5x1)

1.5

2

40

8 (4x2)

4

3

50

6 (2x3)

2

4

70

25.2

6.3

5

100

35

7

6

60

14.4

2.4

7

50

38.5

5.5

8

40

19.2

2.4

9

40

25.2

2.8

10

20

4 (0.4x10)

0.4

µ

µ

å

å

n

n

=

=

=

n

G

(

z

)

P

(

lossj

nL

)

z

P

(

n

defaults

)

z

j

j

=

=

n

0

n

0

m

-

m

n

e

j

µ

n

å

j

n

-

m

+

m

n

z

j

=

=

z

e

G

(

z

)

j

j

j

j

n

!

=

n

0

CreditRisk+: Loss distribution

To derive the distribution of losses for the entire portfolio we proceed as follows:

Step 1: Probability generating function for each band.

Each band is viewed as a portfolio of exposures by itself. The probability generating function for any band, say band j, is by definition:

where the losses are expressed in the unit L of exposure.

Since we have assumed that the number of defaults follows a Poisson distribution (see expression 30) then:

m

m

n

-

m

+

m

å

å

z

j

m

n

j

j

-

m

+

m

z

=

Õ

=

G

(

z

)

e

e

j

=

=

j

j

j

1

j

1

=

1

j

CreditRisk+: Loss distribution

Step 2: Probability generating function for the entire portfolio.

Since we have assumed that each band is a portfolio of exposures, independent from the other bands, the probability generating function for the entire portfolio is just the product of the probability generating functions for all bands.

m

å

m

denotes the expected number of defaults for the entire portfolio.

m

where

=

j

1

j

=

n

1

d

G

(

z

)

=

=

P

(

loss

of

nL

)

|

for

n

1

,

2

,...

=

z

0

n

n

!

dz

j

-

(

)

(

)

v

m

=

=

=

P

0

loss

G

0

e

e

j

j

CreditRisk+: Loss distribution

Step 3: loss distribution for the entire portfolio

Given the probability generating function (33) it is straightforward to derive the loss distribution, since

these probabilities can be expressed in closed form, and depend only on 2 sets of parameters: ej and nj . (See Credit Suisse 1997 p.26)

e

å

e

(

(

)

)

(

)

å

=

-

j

P

loss

of

nL

P

loss

of

n

v

L

j

n

£

j

:

v

n

j

Duffie-Singleton - Jarrow-Turnbull

### V Reduced Form Approach

Reduced Form Approach
• Reduced form approach uses a Poisson process like environment to describe default.
• Contrary to the structural approach the timing of default takes the bond-holders by surprise. Default is treated as a stopping time with a hazard rate process.
• Reduced form approach is less intuitive than the structural model from an economic standpoint, but its calibration is based on credit spreads that are “observable”.
Reduced Form Approach

Example: a two-year defaultable zero-coupon bond that pays 100 if no default, probability of default , LGD=L=60%. The annual (risk-neutral) risk-free rate process is :

=

r

12

%

=

p

0

.

5

´

+

´

´

1

0

.

94

100

0

.

06

0

.

4

100

=

=

V

86

.

08

11

1

.

12

=

r

8

%

´

+

´

´

0

.

94

100

0

.

06

0

.

4

100

=

=

V

87

.

64

12

1

.

1

=

p

0

.

5

=

r

10

%

2

(

)

(

)

´

´

+

´

´

+

´

´

+

´

´

0

.

5

0

.

94

V

0

.

06

0

.

4

V

0

.

5

0

.

94

V

0

.

06

0

.

4

V

=

V

=

11

11

12

12

77

.

52

0

1

.

08

Reduced Form Approach

“Default-adjusted” interest at the tree nodes is:

100

100

=

-

=

=

-

=

R

1

14

.

1

%

R

1

16

.

2

%

12

11

87

.

64

86

.

08

´

+

´

0

.

5

86

.

08

0

.

5

87

.

64

=

-

=

R

1

12

%

0

77

.

52

In all three cases R is solution of the equation ( ):

D

=

t

1

1

1

[

]

=

-

l

D

+

l

D

-

(

1

t

)

t

(

1

L

)

+

D

+

D

1

R

t

1

r

t

D

+

l

D

r

t

tL

D

=

R

t

-

l

D

+

l

D

-

1

t

t

(

1

L

)

®

+

l

If , then , where is the risk-neutral expected loss rate, which can be interpreted as the spread over the risk-free rate to compensate the investor for the risk of default.

l

D

®

R

r

L

L

t

0

é

é

é

ù

ù

ù

ê

ê

ê

ú

ú

ú

ë

ë

ë

û

û

û

Reduced Form Approach

(

)

l

t

General case:is hazard rate, so that if denotes the time to default, the survival probability at horizon t is

t

t

ò

t

>

=

-

l

Prob

(

t

)

E

exp(

(

s

)

ds

)

0

(

)

l

=

l

E is expectation under risk-neutral measure. For the constant we have:

t

t

>

=

-

l

Prob

(

t

)

E

exp(

t

)

(

)

The probability of default over the interval provided no default has happened until time t is:

+

D

t

,

t

t

<

t

£

+

D

=

l

D

Prob

t

t

t

(

t

)

t

(similar to the example above).

Reduced Form Approach

Corporate curve

(

)

R

,

r

R

t

l

L

R

Treasury curve

r

t

Maturity

Term structure of interest rates

Reduced Form Approach

By modelling the default adjusted rate we can incorporate other factors which affect spreads such as liquidity:

=

+

l

+

R

r

L

l

if there is a shortage of bonds and one can benefit from holding the bond in inventory,

if it becomes difficult to sell the bond.

>

l

0

<

l

0

l

l

Identification problem : how to separate and in . Usually is assumed to be given. Implementations differ with respect to assumptions made regarding default intensity .

L

L

L

l

Reduced Form Approach

How to compute default probabilities and

l

Example. Derive the term structure of implied default probabilities from the term structure of credit spreads (assume L=50%).

Company X

One-year

Maturity

Treasurycurve

one-year

forward credit

t (years)

forward rates

FS t

(%)

(%)

(%)

1

5.52

5.76

0.24

2

6.30

6.74

0.44

3

6.40

7.05

0.65

4

6.56

7.64

1.08

5

6.56

7.71

1.15

6

6.81

8.21

1.40

7

6.81

8,47

1.65

P

t

Reduced Form Approach

Forward

Cumulative

Conditional

probabilities

defauilt

default

Maturity

of default

probabilities

probabilities

(years)

t

p

(%)

(%)

(%)

l

t

t

1

0.48

0.48

0.48

2

0.88

1.36

0.88

3

1.30

2.64

1.28

4

2.16

4.74

2.10

5

2.30

6.93

2.19

6

2.80

9.54

2.61

3.30

12.52

2.99

7

l

=

=

l

´

=

2

.

16

FS

L

1

.

08

%

For example, for year 4: , then

4

4

4

(

)

Cumulative probability:

=

+

-

´

l

=

P

P

1

P

4

.

74

4

3

3

4

(

)

Conditional probability:

=

-

´

l

=

p

1

P

2

.

10

4

3

4

Reduced Form Approach
• Generalizations:
• Intensity of the default is modeled as a Cox process (CIR model), conditional on vector of state variables , such as default free interest rates, stock market indices, etc.
• where is a standard Brownian motion, is the long-run mean of is mean rate of reversion to the long-run mean, is a volatility coefficient.
• Properties:
• ,
• Conditional survival probability , where
• and are known time-dependent functions of time,
• The volatility of is

(

)

l

t

(

)

X

t

(

)

(

(

)

)

(

)

(

)

l

=

q

-

l

+

l

s

d

t

k

t

dt

t

dB

t

,

(

)

q

l

B

t

s

k

(

)

l

³

t

0

(

)

(

)

(

)

(

)

a

-

+

b

-

l

=

t

s

t

s

t

p

t

,

s

e

b

a

(

)

(

)

(

)

(

)

=

b

-

s

l

p

t

,

s

s

t

t

p

t

,

s

l

(b.p.)

0

Years

Reduced Form Approach
• Generalizations:
• Intensity of the default can be modeled as a jump process:
• where , - cumulative jumps by at Poisson arrival times, is mean arrival rate, is mean jump size.

(

)

l

t

(

)

(

(

)

)

(

)

l

=

q

-

l

+

d

t

k

t

dt

dZ

t

,

(

)

(

)

(

)

t

=

-

g

Z

t

N

t

Jt

N

t

g

J

Take jumps sizes to be, say, independent and exponentially distributed.

Reduced Form Approach
• Generalizations:
• Risk free spot rate is modeled as one factor extended Vasicek process.
• where , , are similar to parameters in CIR model,
• is a function defined from current term structure of interest rates. is correlated with Brownian motion of the default intensity process.
• Closed form solutions for the bond prices.

(

)

r

t

(

)

(

(

)

(

)

)

(

)

=

q

-

+

s

dr

t

k

t

r

t

dt

dB

t

,

1

1

1

1

(

)

s

B

t

k

1

1

1

(

)

q

t

1

(

)

B

t

1

Reduced Form Approach
• Inputs :
• the term structure of default-free rates
• the term structure of credit spreads for each credit category
• the loss rate for each credit category
• Model assumptions :
• zero correlations between credit events and interest rates
• deterministic credit spreads as long as there are no credit events
• constant recovery rates